general form
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1: 15.11 Riemann’s Differential Equation
2: 9.17 Methods of Computation
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►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979).
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3: 9.16 Physical Applications
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►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation).
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4: 8.20 Asymptotic Expansions of
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§8.20(i) Large
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8.20.1
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8.20.2
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►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii).
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§8.20(ii) Large
…5: 31.11 Expansions in Series of Hypergeometric Functions
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§31.11(ii) General Form
…6: 19.31 Probability Distributions
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and occur as the expectation values, relative to a normal probability distribution in or , of the square root or reciprocal square root of a quadratic form.
More generally, let () and () be real positive-definite matrices with rows and columns, and let be the eigenvalues of .
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►For (19.31.2) and generalizations see Carlson (1972b).
7: Bibliography G
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General form of the quantum-defect theory.
Phys. Rev. A 19 (4), pp. 1485–1509.
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8: 3.6 Linear Difference Equations
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►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6).
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